Asymmetry in Stock Returns: An Entropy Measure Abstract In this paper, we provide an entropy measure for asymmetric comovements between an asset return and the market return. This measure yields a model-free test for asymmetry in stock returns that has greater power than the correlation-based test proposed by Hong, Tu, and Zhou (2007). Based on this test, we find that asymmetry is much more pervasive than previously thought. In the cross-section of stock returns, we find a risk premium (discount) for stocks with high downside (upside) comovement with the market. Asymmetric comovement leaned toward the downside also earns a premium. The risk premia associated with downside comovement. Moreover, downside comovement premium is almost twice as large as the premium of downside beta. Our findings are consistent with theoretical implications of a representative agent model with disappointment aversion preferences. Keywords: Asymmetric dependence, metric entropy, asset pricing, anomaly. JEL Classification: C12, C15, C32, G11, G12, G17 1 1 Introduction Asymmetric characteristics of asset returns, i.e. stocks co-move more strongly when market goes down than when market goes up, have been found by a number of prior studies. Ball and Kothari (1989); Schwert (1989); Conrad, Gultekin, and Kaul (1991); Cho and Engle (1999); Bekaert and Wu (2000); Ang and Chen (2002); Bae, Karolyi, and Stulz (2003); Ang, Chen, and Xing (2006), among others, documented asymmetries in covariances, cor- relations, volatilities, and betas of stock returns. Such asymmetric characteristics of stock returns are important both for portfolio selection and risk management, because effective hedging relies on the dependence between assets hedged and financial instruments used. If the dependence structure is varying with the state of the market, portfolio managers may need to worry about the effectiveness of their hedges when they are most needed. Almost all these previous studies are model dependent. For example, Ang and Chen (2002) find correlation asymmetries among various portfolios under joint normality as- sumption, which leaves the room for possibility that the data features unexplained by the joint normal model may well be explained by some other symmetric model. Hong, Tu, and Zhou (2007) proposes the first and the only model-free test of asymmetry to date. This test, despite of its novelty, has two weaknesses. First, it detects only asymmetric correla- tions, and does not address asymmetry beyond the second moment. It is well known that the correlation coefficient is only a measure of linear dependence and cannot capture the full dependence structure for non-normal distributions, while several papers documented that realized stock returns are non-normally distributed (see, e.g., Embrechts, McNeil, and Straumann, 2002; Ang and Chen, 2002). Second, its finite-sample power seems low in empirical applications. For example, the test cannot detect any asymmetry in portfolios sorted by book-to-market ratio, and finds only one significant asymmetric portfolio among momentum sorted portfolios. In this paper, we first propose an entropy measure to exactly measure the asymmetric dependence between individual stock return and the market return. Using the entropy measure, we propose a new model-free test for asymmetric dependence. The test statistic 2 is a normalized metric entropy proposed by Granger, Maasoumi, and Racine (2004) that have been widely applied in econometrics (see, e.g., Maasoumi and Racine, 2002; Racine and Maasoumi, 2007). The entropy measure is defined based on the joint probability density functions, so it can summarize all the information of a given joint distribution and hence can capture general asymmetric dependence structure existed in all the moments. With Monte Carlo simulations, we find that the entropy-based test has good size and power properties. Using sorted portfolios based on size, book-to-market ratio and momentum, the entropy-based test detects statistically significant asymmetry in all kinds of the sorted portfolios. For example, in contrast to the Hong, Tu, and Zhou (2007) test, we find asymmetry in 2 portfolios at the 5% significance level, and in 7 portfolios at the 10% level, out of the 10 decile portfolios sorted by the book-to-market ratio. What is the asset pricing implication of asymmetry in the cross-section of expected stock returns? It is actually a less studied question in the literature. Under classical Capital Asset Pricing Model (CAPM), it is sufficient to consider only linear correlations (captured by the CAPM beta) between individual stock returns and the market portfolio return. (see Sharpe, 1964; Lintner, 1965). However, more recent studies find supporting evidence that asymmetry features of the joint distribution of individual stock and market returns also determine the expected stock returns. For example, Harvey and Siddique (2000) show that the conditional coskewness plays an important role in explaining the cross-sectional returns. Ang, Chen, and Xing (2006) find that asymmetric risk premia are associated with downside and upside betas. They show that stocks with higher downside betas have on average higher returns, but have mixed evidence on whether higher upside betas are associated with lower returns. Since downside and upside betas are highly correlated with market betas (the correlations are above 0.75 as shown in Ang, Chen, and Xing (2006)), i.e. an increase in downside or upside betas are associated with an increase in the CAPM beta, it is difficult to distinguish the effects of downside or upside covariation from the overall covariation between the stock and the market returns. Alcock and Hatherley (2013) tries to overcome this problem by constructing a beta-invariant asymmetric dependence measure that is a 3 modified J statistic proposed by Hong, Tu, and Zhou (2007). Although beta-invariant, their measure still does not capture full dependence structure since it is constructed based on exceedance correlations that can only capture conditional dependence to the second moment (linear dependence). Some recent papers start to examine cross-sectional asset pricing implications of higher order dependence. Contemporary paper by Chabi-Yo, Ruenzi, and Weigert (2014), in a non-normal distribution framework, uses parametric copula-based tail dependence measure to explain the cross-sectional expected returns. This paper differs from theirs in that they focus on extreme lower tail dependence, or the crash sensitivities of stocks, while, muck like Ang, Chen, and Xing (2006), we focus on the downside and upside dependence when mar- ket returns are above or below the mean. Using the proposed entropy measure, we study the asset pricing implications of asymmetric dependence. Theoretically, Ang, Chen, and Xing (2006) shows that under a simple representative agent model with disappointment aversion (DA) utility (Gul, 1991), agents require a premium to hold stocks with strong covariation with the downside market, while are willing to hold stocks with high upside potential at a discount, all else being equal. Motivated from this insight, we expect stocks with stronger downside asymmetric dependence, i.e. the dependence with the downside market is stronger than with the upside market, to earn higher average returns, because those stocks are highly risky in the sense that they may incur large loss when the wealth level is low, meanwhile they do not have high upside potential when the market goes up. Furthermore, as pointed out by Ang, Chen, and Xing (2006), the DA utility is kinked at certainty equivalence wealth level, so the higher-order co-moments derived from Tay- lor expansion, like coskewness and cokurtosis, may not approximate the utility function well globally. This is a theoretical motivation why there may exist asymmetric effects of downside and upside dependence. We construct proxies for downside and upside dependence using estimated probabili- ties that individual stock and market returns both fall below or above the sample means. Using Center for Research in Securities Prices (CRSP) data from 1962 to 2013, we find 4 empirical evidence that stocks with high downside (upside) dependence earn a premium (discount). Both effects are statistically and economically significant after controlling for other known characteristics in cross-sectional Fama and MacBeth (1973) regressions. The findings support the theoretical implications of a representative agent model with DA u- tility. The value-weighted average return (Carhart (1997) four factor adjusted alpha) of the top quintile portfolio sorted based on downside asymmetric dependence outperforms the lowest quintile portfolio by 12.34% (12.89%) per annum. In Fama-Macbeth (1973) re- gressions, the premium of downside asymmetric dependence cannot be explained by known characteristics, such as CAPM beta, downside or upside betas, coskewness and cokurtosis, size, book-to-market ratio, past returns and maximum daily return within a month. The downside asymmetric dependence is time-varying and shows limited predictability using its own lag. Yet when using the lagged asymmetric dependence to form a trading strategy, the spread portfolio still earns an average equal-weighted annualized return of 4.5%. The premium is both economically and statistically significant. The rest of the paper is organized as follows. Section 2 introduces the entropy-based test for asymmetric dependence. Section 3 examines the test size and power using Monte Carlo simulations. Section 4 applies the entropy test to investigate asymmetry